The Phi-Function.
Maths Coursework by Yasir Al-Wakeel The Phi-Function A Function in mathematics is the term used to indicate the relationship or correspondence between two or more quantities. It was first used in 1637 by a French mathematician by the name of Rene Descartes to designate a power xn of a variable x. Later, Gottfried Wilhelm Leibniz in 1694 applied the term to various aspects of a curve, such as its slope. The most widely used meaning until quite recently was defined in 829 by the German mathematician Peter Dirichlet. Dirichlet conceived a function as a variable y, called the dependent variable, having its values fixed or determined in some definite manner by the values assigned to the independent variable x, or to several independent variables x1, x2, ..., xk. The Phi-Function is a means of breaking down numbers. It is defined as the number of positive integers less than n, where n is a positive integer, which are co-prime with n. Zero is neither a positive integer nor a negative integer, it is on the boundary and so does not come under the notation of n. The phi function of a positive inter, n, is expressed as ?(n). Two terms are co-prime when they have no factor in common other than one. For example 3 and 4 are co-prime or 5, 7, and 8 are all co-prime. When numbers are co-prime they can be written: (n , m)=1 such as (5,7)=1.
The totient function.
Maths Coursework The totient function ?(n), also called Euler's totient function, is defined as the number of positive integers <= n which are relatively prime to (i.e., do not contain any factor in common with) n, where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function ? (n) can be simply defined as the number of totatives of n. For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so ?24=8. The totient function is implemented in Mathematics as EulerPhi [n]. In this part of the coursework will be investigating the phi function of: ) ?(p) 2) ?(m x n) = ?(m) x ?(n) 3) ?(pn) 4) Other methods of calculating the Phi values of an integer that I found from the net. Find the value of: (i) ?(3): = 2 3 = 1,2 The number 3 only has 2 positive co-prime integers they are the numbers 1 and 2. (ii) ?(8): = 4 8 = 1,3,5,7 There are 4 positive co-prime integers for the number 8 (iii) ?(11): = 10 1 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive co-prime integers, and they are shown above. (iv) ?(24): = 8 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive co-prime integers, and they are shown above. Part 1 In this part I will try to investigate on ?(n). But before we can start we need to experiment on certain numbers.
Identify and explain the rules and equations associated with the Phi function.
Introduction The Phi is described as the number of positive integers less than n (positive integer) which have no factor, other than 1, in common, co-prime, with n. In my investigation I aim to identify and explain the rules and equations associated with the Phi function. To go about this I will investigate ? (n). This will be further touched upon and will help me investigate and support any conclusions I hope to gain from investigating whether (a x b) = (a) x (b) in certain cases. Table of Phi's from 2-40 Number Phi ?2 , ?3 ,2 2 ?4 ,2,3 2 ?5 ,2,3,4 4 ?6 ,2,3,4,5 2 ?7 ,2,3,4,5,6 6 ?8 ,2,3,4,5,6,7 4 ?9 ,2,3,4,5,6,7,8 6 ?10 ,2,3,4,5,6,7,8,9 4 ?11 ,2,3,4,5,6,7,8,9,10 0 ?12 ,2,3,4,5,6,7,8,9,10,11 4 ?13 ,2,3,4,5,6,7,8,9,10,11,12 2 ?14 ,2,3,4,5,6,7,8,9,10,11,12,13 6 ?15 ,2,3,4,5,6,7,8,9,10,11,12,13,14 8 ?16 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15 8 ?17 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 6 ?18 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 6 ?19 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18 8 ?20 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 8 ?21 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 2 ?22 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21 0 ?23 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21,22 22 ?24 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21,22,23 8 ?25
Literary techniques.
Words create mood and context, and for this purpose old-sounding, old-fashioned Imagery is the content of thought where attention is directed to sensory qualities - i.e. mental images, figures of speech and embodiments of non-discursive truth. Metaphor commonly means saying one thing while intending another, making implicit comparisons between things linked by a common feature, perhaps even violating semantic rules. alliteration, assonance, euphony, rhyme, pararhyme, onomatopoeia, repetition and tone colour. The opening of a narrative is important because it usually sets the scene and gives clues as to what the story will be about. Openings are important because they need to grab the audience's attention. People may walk out or turn over if the film has a bad opening or a slow one. The opening is the narrative hook to get the audience to stay and watch. The opening of a text is usually defines as more than the first paragraph but even in such a small unit it is possible to see some narrative features necessary to get us into the story. In a carefully constructed narrative, the opening will contain the essence of the themes that will later develop. When an audience begins reading a text they engage in the process of predicting what will happen based on the clues give. Most films want to orientate the audience quickly so they give them unambiguous signs. A sign's
The Phi function.
Maths Coursework The Phi function The phi function says that for any positive integer such as n the phi is ?(n) and is defined as the number of positive integers, less than n which have no factor (other than 1) in common which means that they are co prime with n. If we take for example the phi of various numbers we will find that there is some relationship between them and I will investigate this relationship throughout. I will also be investigating the results and the formulae for ?(n2) and thus the formula for ?(nx). For example ?(20) = 8 This was obtained by the following ways First, list all the factors of 20 and all the numbers till 20. ,2,4,5,10,20 These numbers can immediately be cancelled from the list of numbers till 20. The remaining numbers are 3,6,7,8,9,11,12,13,14,15,16,17,18,19. But the numbers 6,8,12,14,15,16,18 can also be cancelled out because one of their factors is the same as the factor for 20 and so cannot be considered. So from the following list ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 and so the final numbers remaining are 1,3,7,9,11,13,17 and 19 and so ?(20) = 8 This way we can draw a table as is shown on the next page to find out the ? values for all the numbers till 40. The table is shown on the next page. By using this table we will investigate further on the relationships of the values. We will find the value of various numbers
Millikan's theory.
Representations are for Millikan part of a larger group of entities for which she considers there is no generic name in English, and which would include "[n]atural signs, animals' signs, people's signs, indexes, signals, indicators, symbols, representations, sentences, maps, charts, pictures" [LTOBC, 85]. For want of a better term, Millikan calls these entities signs, and claims that what is common to all signs is their being, to a greater or lesser degree, intentional. That is, what all signs have in common, in a family resemblance way, is their bearing a certain relationship to entities other than themselves - a relationship which is usually characterized as "being about something else", "meaning something else". In what follows, I will consider what the nature of this "being about something else" is according to Millikan. I will pay particular attention to mental, or inner, representations, despite the fact that Millikan believes "articulate conventional signs" - that is, I take it, verbal utterances - to be the paradigm case of signs. For, like Searle, Millikan regards verbal utterances and other external verbal-like modes of representation (such as writing), to have an intentionality derived from the original intentionality of states of mind, and thus explainable in terms of the latter. Intentionality is, according to Millikan, a question of degree: indeed, she rejects
Maths Primes and Multiples Investigation
. A)I) ?(3)-1, 2=2 II) ?(8)-1, 2, 3, 4, 5, 6, 7=4 III) ?(11)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10=10 IV) ?(24)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23=8 B)I) ?(5)-1, 2, 3, 4=4 II) ?(10)- 1, 2, 3, 4, 5, 6, 7, 8, 9=4 III) ?(15)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14=8 IV) ?(20)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19=8 V) ?(50)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49=20 2. A)I) ?(7x4) = ?(7) x ?(4) 7x4=28 ?(28)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27=12 ?(7)=1, 2, 3, 4, 5, 6=6 ?(4)=1, 2, 3=2 2x6=12 12=12, therefore a prime and an even work(non-prime). B) ?(6x4) = ?(6) x ?(4) 6x4=24 ?(24)=8 ?(6)=1, 2, 3, 4, 5=2 ?(4)=2 2x2=4 4=8, therefore two evens don't work. C) ?(5x10)= ?(50) ?(50)=20 ?(10) x ?(5) 4x4=16 16=20, therefore two multiples don't work. ?(13x3)= ?(39) ?(39)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38=24 ?(13)=12 ?(3)=2 2x12=24 24=24, therefore two primes work. Having done these examples I below have drawn up a table of my results. When I made the table I saw
In this coursework I was asked to investigate the Phi Function (f) of a number (n).
In this coursework I was asked to investigate the Phi Function (?) of a number (n). The Phi Function of a number (n) is delineated as the number of positive integers less than n, which have no factor (other than 1) in common, i.e. co-prime with n. Example: ?(16) = 8. The integers less than 16 that have no factors apart from 1 in common with 16 are 1, 3, 5, 7, 9, 11, 13, and 15. There are 8 altogether. To calculate ?(n) I will list out the numbers from 1 till n?1. I will then cross out all the numbers that have a common factor with n. The remaining numbers will give me the ? of n. In this part of the coursework will be investigating the phi function of: ) ?(p) 2) ?(p)² Part 1: Find the value of: (I) ?(3): 1 2 3 = 1,2 The number 3 only has 2 positive co-prime integers they are the numbers 1 and 2. (ii) ?(8): 2 3 4 5 6 7 8 = 1,3,5,7 There are 4 positive co-prime integers for the number 8 (iii) ?(11): 2 3 4 5 6 7 8 9 10 11 1 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive co-prime integers, they are shown above. (iv) ?(24): 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive co-prime integers, they are shown above. I will create a table for the Phi values of the numbers from 1 to 30. I have created this table because it would be easier to lookup the phi values of the numbers within this
The Phi Function Investigation
The Phi Function For any positive integer n, the Phi Function ?(n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n. Part 1 (a) Find the value of: (I) ?(3) (ii) ?(8) (iii) ?(11) (iv) ?(24) (b) Obtain the Phi-Function for at least 5 positive integers of your own choice. (a) (I) ?(3): 1 2 1,2 3 1,3 3 = 1,2 The number 3 only has 2 positive integers they are the numbers 1 and 2. (ii) ?(8): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 8 = 1,3,5,7 There are 4 positive integers for the number 8 (iii) ?(11): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 0 1,2,5,10 1 1,11 1 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive integers, they are shown above. (iv) ?(24): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 0 1,2,5,10 1 1,11 2 1,2,4,6,12 3 1,13 4 1,2,7,14 5 1,3,5,15 6 1,2,4,8,16 7 1,17 8 1,2,6,9,18 9 1,19 20 1,2,4,5,10,20 21 1,3,21 22 1,2,11,22 23 1,23 24 1,2,3,4,6,8,12,24 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive integers, they are shown above. (b) Obtain the Phi-Function for at least 5 positive integers of your own choice. (I) ?(6): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 6 = 1 and 5 The
Investigating the Phi function
Maths coursework Phi function Investigating the Phi function The phi function is defined for any positive integer(n), as the number of positive integers not greater than and co-prime (have no factor other than 1 in common) to n Example So (12) = 4 because the integers less than 12 which have no factors in common with it except for 1 are 1,5,7,11 i.e. there is 4 of them. I started to investigate the phi function of numbers from 2 to 24 so I could find patterns, which I can use to create a formula for the(n) term (n) Shared factors Not sharing factors (2) - (2) = 1 (3) ,2 (3) = 2 (4) 2 ,3 (4) = 2 (5) ,2,3,4 (5) = 4 (6) 2,3,4 ,5 (6) = 2 (7) ,2,3,4,5,6 (7) = 6 (8) 2,4,6 ,3,5,7 (8) = 4 (9) 3,6 ,2,4,5,7,8 (9) = 6 (10) 2,4,6,8,5 ,3,7,9 (10) = 4 (11) ,2,3,4,5,6,7,8,9,10 (11) = 10 (12) 2,4,6,8,10,3,9 ,5,7,11 (12) = 4 (13) ,2,3,4,5,6,7,8,9,10,11,12, (13) = 12 (14) 2,4,6,8,10,12,7 ,3,5,11,13 (14) = 6 (15) 3,5,9,12,6,10 ,2,4,7,8,11,13,14 (15) = 8 (16) 2,4,6,8,10,12,14 ,3,5,7,11,13,15 (16) = 8 (17) ,2,3,4,5,6,7,8,9,10,11,12,13, 4,15,16 (17) = 16 (18) 2,3,4,6,8,10,12,14 ,5,7,11,13,17, (18) = 6 (19) ,2,3,4,5,6,7,8,910,11,12,13,14, 5,16,17,18 (19) = 18 (20) 2,4,6,8,10,12,1,4,16,18,5,15 ,2,4,5,8,10,11,13,16,17,19 (20) = 8 (21) 3,6,9,12,15,18,7,14 ,2,4,5,8,10,11,13,16,17,19,20 (21) = 12 (22)